Abstract

We prove that the renormalized defocusing modified KdV (mKdV) equation on the circle is locally in time $C^{0}$-well-posed on the Fourier Lebesgue space ${\mathscr{F}\ell}^p$ for any $2 < p < \infty$. The result implies that the defocusing mKdV equation itself is ill-posed on these spaces since the renormalizing phase factor becomes infinite. The proof is based on the fact that the mKdV equation is an integrable PDE whose Hamiltonian is in the nonlinear Schrödinger hierarchy. A key ingredient is a novel way of representing the bi-infinite sequence of frequencies of the renormalized defocusing mKdV equation, allowing us to analytically extend them to ${\mathscr{F}\ell}^p$ for any $2 \le p < \infty$ and to deduce asymptotics for $n \to \pm \infty$.

Keywords

  1. modified KdV equation
  2. frequency map
  3. well-posedness
  4. ill-posedness

MSC codes

  1. Primary
  2. 37K10; Secondary
  3. 35Q53
  4. 35D05

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Information & Authors

Information

Published In

cover image SIAM Journal on Mathematical Analysis
SIAM Journal on Mathematical Analysis
Pages: 2191 - 2219
ISSN (online): 1095-7154

History

Submitted: 3 October 2016
Accepted: 16 March 2017
Published online: 22 June 2017

Keywords

  1. modified KdV equation
  2. frequency map
  3. well-posedness
  4. ill-posedness

MSC codes

  1. Primary
  2. 37K10; Secondary
  3. 35Q53
  4. 35D05

Authors

Affiliations

Funding Information

Swiss National Science Foundation

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